UC Santa Barbara

This dissertation discusses various quantum integrable structures underlying gauge the-ories. We demonstrate that such structures naturally emerge when we connect the gauge theories to string theory by taking the large N limit of the color degrees of freedom. Such integrable structures are not only tools for exact computations for various physical quantities in gauge theory, but also give guidance to further elucidate the correspondence between gauge theory and string theory.We first discuss the integrable spin chains appearing in the weak 't Hooft coupling expansion of the N4 super Yang-Mills theory, which has been well known to be dual to type IIB string theory on the AdS5 x S geometry. We specifically focus on the boundary conditions induced by the Gukov-Witten-type superconformal surface defects, which happen to be integrable at some special points in the defect moduli space but not for generic configurations. Additionally, aspects of the surface defects are further studied by taking advantage of the superconformal symmetry for future investigation of their (non-)integrability.The second part of the dissertation discusses the spin chain structures emerging in the confining string sectors of large N lattice Yang-Mills theory. The nontriviality of the confining string theory, distinct from the ordinary Nambu-Goto theory, is due to the unitarity constraint, or the so-called zigzag constraint, that is naturally required for gauge theory due to the gauge holonomy being a unitary group. We indeed find that the spin chains for certain closed subsectors are integrable when the unitarity constraint is trivial, but the integrability is violated generally due to the constraint. We also demonstrate that the integrability can be utilized to study the physics of the gauge theory. In particular, it successfully estimates the roughening transition point from the energy spectra exactly computed using the Bethe ansatz method.